| L(s) = 1 | + 2·2-s + 10·3-s + 4·4-s + 20·6-s + 8·8-s + 73·9-s + 9·11-s + 40·12-s − 52·13-s + 16·16-s + 96·17-s + 146·18-s + 10·19-s + 18·22-s − 75·23-s + 80·24-s − 104·26-s + 460·27-s + 189·29-s + 232·31-s + 32·32-s + 90·33-s + 192·34-s + 292·36-s − 305·37-s + 20·38-s − 520·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.92·3-s + 1/2·4-s + 1.36·6-s + 0.353·8-s + 2.70·9-s + 0.246·11-s + 0.962·12-s − 1.10·13-s + 1/4·16-s + 1.36·17-s + 1.91·18-s + 0.120·19-s + 0.174·22-s − 0.679·23-s + 0.680·24-s − 0.784·26-s + 3.27·27-s + 1.21·29-s + 1.34·31-s + 0.176·32-s + 0.474·33-s + 0.968·34-s + 1.35·36-s − 1.35·37-s + 0.0853·38-s − 2.13·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.806300097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.806300097\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 96 T + p^{3} T^{2} \) |
| 19 | \( 1 - 10 T + p^{3} T^{2} \) |
| 23 | \( 1 + 75 T + p^{3} T^{2} \) |
| 29 | \( 1 - 189 T + p^{3} T^{2} \) |
| 31 | \( 1 - 232 T + p^{3} T^{2} \) |
| 37 | \( 1 + 305 T + p^{3} T^{2} \) |
| 41 | \( 1 - 438 T + p^{3} T^{2} \) |
| 43 | \( 1 + 353 T + p^{3} T^{2} \) |
| 47 | \( 1 + 486 T + p^{3} T^{2} \) |
| 53 | \( 1 - 354 T + p^{3} T^{2} \) |
| 59 | \( 1 - 672 T + p^{3} T^{2} \) |
| 61 | \( 1 + 206 T + p^{3} T^{2} \) |
| 67 | \( 1 + 599 T + p^{3} T^{2} \) |
| 71 | \( 1 + 471 T + p^{3} T^{2} \) |
| 73 | \( 1 - 614 T + p^{3} T^{2} \) |
| 79 | \( 1 - 743 T + p^{3} T^{2} \) |
| 83 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 89 | \( 1 + 180 T + p^{3} T^{2} \) |
| 97 | \( 1 + 184 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.355087888696882393949025831023, −7.937032284457733750527799054871, −7.18871558385780130550565802205, −6.47224538644911149888540999941, −5.17042793005422463812789927740, −4.41489418899532732030168522596, −3.56251018438698402249423040913, −2.90935059093269541453871202917, −2.17838050957176120973729379505, −1.15806321342478619505383498047,
1.15806321342478619505383498047, 2.17838050957176120973729379505, 2.90935059093269541453871202917, 3.56251018438698402249423040913, 4.41489418899532732030168522596, 5.17042793005422463812789927740, 6.47224538644911149888540999941, 7.18871558385780130550565802205, 7.937032284457733750527799054871, 8.355087888696882393949025831023
